An oscillating classical background can lead to particle production [1]}, [2]}, [3]}. After inflation, \(\phi \)  oscillates coherently in either \(\phi ^2\)  or \(\phi ^4\)  potential, depending on the inflationary model.
As a result, the \(\phi -s\)  couplings induce dark matter pair production. If the corresponding coupling is small, the process can be described perturbatively. Below, we consider the
contributions of the 3 basic operators to this reaction. We treat the Hubble expansion adiabatically such that the time dependence can be inserted in the inflaton oscillation amplitude at the end of the calculation. Also, we treat the produced dark matter particles as free and neglect backreaction. These approximations are justified at small inflaton–DM couplings.
